Farnes is a postdoc fellow at the Oxford e-Research center and has previously worked on observational astrophysics. A few days ago, Oxford University published a press-release celebrating the publication of Farnes’ paper. This press-release was then picked up by phys.org and spread from there to a few other outlets. I have since gotten various inquiries by readers and journalists asking for comments.

In his paper, Farnes has a go at cosmology with negative gravitational masses. He wants these masses further to also have negative inertial masses, so that the equivalence principle is maintained. It’s a nice idea. I, as I am sure many other people in the field, have toyed with it. Problem is, it works really badly.

General Relativity is a wonderful theory. It tells you how masses move under the pull of gravity. You do not get to choose how they move; it follows from Einstein’s equations. These equations tell you that like masses attract and unlike masses repel. We don’t normally talk about this because for all we know there are no negative gravitational masses, but you can see what happens in the Newtonian limit. It’s the same as for the electromagnetic force, just with electric charges exchanged for masses, and – importantly – with a flipped sign.

The deeper reason for this is that the gravitational interaction is exchanged by a spin-2 field, whereas the electromagnetic force is exchanged by a spin-1 field. Note that for this to be the case, you do not need to speak about the messenger particle that is associated with the force if you quantize it (gravitons or photons). It’s simply a statement about the type of interaction, not about the quantization. Again, you don’t get to choose this behavior. Once you work with General Relativity, you are stuck with the spin-2 field and you conclude: like charges attract and unlike charges repel.

Farnes in his paper instead wants negative gravitational masses to mutually repel each other. But general relativity won’t let you do this. He notices that in section 2.3.3. where he goes on about the “counterintuitive” finding that the negative masses don’t actually seem to mutually repel.

He doesn’t say in his paper how he did the N-body simulation in which the negative mass particles mutually repel (you can tell they do just by looking at the images). Some inquiry by email revealed that he does not actually derive the Newtonian limit from the field equations, he just encodes the repulsive interaction the way he thinks it should be.

Farnes also introduces a creation term for the negative masses so he gets something akin dark energy. A creation term is basically a magic fix by which you can explain everything and anything. Once you have that, you can either go and postulate an equation of motion that is consistent with the constant creation (or whatever else you want), or you don’t, in which case you just violate energy conservation. Either way, it doesn’t explain anything. And if you are okay with introducing fancy fluids with uncommon equations of motion you may as well stick with dark energy and dark matter.

There’s a more general point to be made here. The primary reason that we use dark matter and dark energy to explain cosmological observations is that they are simple. Occam’s razor vetoes any explanation you can come up with that is more complicated than that, and Farnes’ approach certainly is not a simple explanation. Furthermore, while it is okay to introduce negative gravitational masses, it’s highly problematic to introduce negative inertial masses because this means the vacuum becomes unstable. If you do this, you can produce particle pairs from a net energy of zero in infinitely large amounts. This fits badly with our observations.

Now, look. It may be that what I am saying is wrong. Maybe the Newtonian limit is more complicated that it seems. Maybe gravity is not a spin-2 interaction. Maybe you can have mutually repulsive negative masses in general relativity after all. I would totally be in favor of that, as I have written a paper about repulsive gravity myself (it’s quoted in Farnes’ paper). I believe that negative gravitational masses are the only known solution to the (real) cosmological constant problem. But any approach that attempts to work with negative masses needs to explain how it overcomes the above mentioned problems. Farnes’ paper falls short of this.

In summary, the solution proposed by Farnes creates more problems than it solves.

If we give all fundamental units of matter a magnetic moment due to spinning mass, gravitational repulsion would be similar to electromagnetic repulsion at close range. Furthermore, righthanded neutrinos would not be super massive, but antineutrinos.

I second your assessment of this paper. Farnes violates the weak energy condition with T^{00} < 0 in this fluid that then generates positive mass. This sounds a bit like Hoyle's old idea of creation fields in steady state cosmology. This also runs into issues with chronology and thermodynamics. I have not looked at all into the astrophysics of this, but getting halos of negative matter to maintain stable galaxies and clusters seems like a delicate balancing act.

"Thank you for writing an article about this. However, I do not think these comments are actually related to the findings in my paper, but rather the papers of others. Your disagreement appears to be with the work of Bondi, who showed that these negative masses are compatible with GR. The comments seem to ignore Bondi's seminal work. I highlight in my paper that spin-2 particles are not at all relevant in this model - I know that is the lens through which you view these equations, but it is just one of many perspectives.

There also seems to be some confusion about section 2.3.3. and the “counterintuitive” finding. This is not actually related to my own work, but is actually an outcome from the cited work of Stephen Hawking and Don Page. It's not counterintuitive because it is wrong, just because that is how the predicted universe would behave!

So the article in its current form gives the impression that it disagrees with my paper, but you are actually disagreeing with far more influential works and authors.

A creation term is also not a "magic fix by which you can explain everything and anything". That is incredibly misleading. It provides very exact and specific well-defined physical properties.

The article also currently reads: "The primary reason that we use dark matter and dark energy to explain cosmological observations is that they are simple." Here you are neglecting the fact that there is ***no physical explanation*** for either dark energy and dark matter. My theory provides the first physical explanation for both of these phenomena in a single unified and intuitive framework. Given the lack of evidence for all conventional theories at present, including those which you frequently highlight, I am surprised that you would not see the advantage to a new idea.

Highlighting the vacuum instability is also completely wrong. This is a feature of the theory, and this is clearly emphasised in the paper. The creation term moderates the production rate of negative mass particles, and prevents this from being a catastrophic event.

Having said all that, I do greatly appreciate the last paragraph, which I think is much more correct. The article also does not mention the abundance of astrophysical observations which my model seems to match - I think to present it as a theory is not really fair or accurate, given the initial matching to the real world."

The "counterintuitive" finding that you write about is that "although the negative massesare gravitationally repelling one another, the cosmological effect appears to be for the negative energy associated with negative masses to cause the universe to recollapse."

That's because they're not repelling each other. If you think that Hawking & Page said they did, you must have misread the paper.

You also apparently didn't think about what I wrote about the spin-2 field. Not particle. I suggest you do.

You have some confusion about what "explanation" means.

"Given the lack of evidence for all conventional theories at present, including those which you frequently highlight, I am surprised that you would not see the advantage to a new idea."

It would be an advantage if it worked. But it does not.

Look, I understand that this must be annoying for you. Trust me, I do not enjoy doing this, but I do not want false claims to spread in the popular science literature. If you do not want to take it from me, go and ask anyone who has worked on modifications of gravity. They will almost certainly tell you the same.

Hi Sabine, Thanks for your insights. I have been interested in the possibility of negative gravitational masses for quite a time now, and it seems that there's still a lot of confusion, be it in my mind or in the scientific community. First, there is something you say that is deeply confusing. How can General Relativity (GR) be compatible with like charge attraction and unlike charge repulsion? This would immediately imply that there's a violation of both (weak and strong) equivalence principles, which are fundamental to the validity of GR. Let's say we bring a negative gravitational mass near Earth gravitational field: because it's gravitational mass is negative and it's inertial mass positive it will be repeled by it, whereas a positive gravitational mass will be attracted by it. GR equations can't do that, because the space time curvature is imposed by the Earth field, it is thus "positive", and every type of mass should thus fall! It seems to me that Bondi type negative masses are more compatible with GR since they don't violate the equivalence principle, even if other problems arise. Let's say for instance that alpha G experiment observes antigravity, that is hydrogen anti-atoms falling up with -g acceleration: in my mind it is clear that this would falsify GR and put an end to the idea of space time curvature as a physical explanation of gravitational phenomena, won't it? Thanking you in advance for your answers!

Right, in GR any type of mass will fall towards a positive mass object, including negative masses. Jamie assumes that in his paper. Note that this uses a test-mass approximation (geodesic equation). I believe that one of the papers he cites (about the "runaway" problem) deals with the transition.

"The primary reason that we use dark matter and dark energy to explain cosmological observations is that they are simple. Occam’s razor vetoes any explanation you can come up with that is more complicated than that, and Farnes’ approach certainly is not a simple explanation."

I think his approach is more parsimonous than dark matter and dark energy, since it explains the observations using a single dark fluid, instead of two completely unconnected dark things. It minimizes the number of unproven entities, even if it does not eliminate them.

Whether it fails due to other things you pointed out (general relativity problems with negative inertial masses), I am not qualified to comment on. But if it does not, I think it would a preferrable explanation to current hypotheses.

I'm a bit confused. You say Jamie is using particles with both negative gravitational and inertial mass, but that he wrongly assumes they repel each other. Isn't he right? In the Newtonian limit, the force between a pair of such particles will be the same as for a pair of positive mass particles (two flipped signs) i.e. towards each other. But then their accelerations will be away from each other because of the additional sign in F=ma. No?

It does not, please look at the paper. You need to introduce some weird stuff new stuff, then you need to introduce the creation tensor, then you need to assume you have no problem with vacuum stability, then you need to somehow assume that you get around the issue with the spin-2 field while still using GR, then you need to explain how come that a negative cosmological constant is actually in agreement with all the data, and even if you have done that you'd still have to bend over backward to demonstrate that the solution actually does fit the rotation curves which, frankly, I am rather skeptic about because I cannot see how you get the right scaling behavior (Tully-Fisher and all). Best,

Interesting article. In summer 2018 I wrote two papers on vixra. The papers used Newton's Laws, rather than GR, to simulate, by computer, the motions of an aggregate of some positive-mass particles and some negative-mass particles. My findings were that negative mass, in a naive/broad interpretation, mimics simultaneously both dark energy and dark mass.

The first paper referred to negative mass particles (with other particle properties unspecified) while the second paper suggested "what if" the negative mass particles were antiparticles. Both types of negative masses were met in an inline forun by a large dose of disbelief, particularly the latter type. I accept the argument that if antimatter had negative mass then it would contradict the Feynman-Stueckelberg method which underlies Feynman diagrams. However it is not the first time that a sign flip has caused problems, so I am still (in early stages of) trying to think how to circumvent this objection ...

One point is that can GR only be taken to be known to work in a region where positive mass dominates?

Second, all masses (both + and -) are gravitationally attracted to positive masses (which are almost every known mass), so how can we distinguish positive from negative masses in their gravitational behaviour in our matter-dominated universe which attracts all masses?

Third, is there a possibility that the energy liberated by annihilating a negative mass is also negative energy. And what is the possibility that this particular negative energy also appears to be positive energy when measured in our matter-dominated universe?

Since you mentioned Occam’s razor in your argument there is an even simpler explanation science doesn’t seem to yet embrace. That, GR has more in common with Newtonian mechanics than we currently believe; they are both tremendous accomplishments and leaps forward in our understanding of motion and gravity, and they are both not completely correct in how they represent that. What eventually comes after GR will likely be as different as GR was to Newton. I’m likely too simple minded in thinking there is a more basic way to preserve spacetime malleability without gravity physically curving space, and seeing those dynamics are already implicit in SR.

"I believe that negative gravitational masses are the only known solution to the (real) cosmological constant problem".

What cosmological constant problem? I thought it's just an observed fact that the CC isn't exactly zero, and that all data about the expansion of the universe is compatible with a constant CC in General Relativity.

@Sabine: "The deeper reason for this is that the gravitational interaction is exchanged by a spin-2 field, whereas the electromagnetic force is exchanged by a spin-1 field."

Have you ever written about what a spin-2 field is and why it's implicit in the gravitational interaction? I'd be interested in the details. Spin makes some sense to me with regard to particles, but I don't understand it with regard to fields.

Looking closely on the "section 2.3.3 disagreement" between Sabine and Jamie.

At the beginning of the section 2.3.3 Jamie writes:"I showed in Sect. 2.3.2 that Λ can be modelled as a constant density of negative masses, which remain constant via matter creation."

Now, Hawking&Page 1982 paper about black holes in AdS space with negative Λ speaks only about negative energy density, not mass. Then, here is the question: since there is mass-energy equivalence does negative energy (density) imply automatically negative mass?

Am I right that this sentence from Hawking&Page paper suggests it's not? "The positive mass theorem for anti-de Sitter space [13, 14] indicates that there should not be any states for negative energies." This theorem is mentioned earlier to be extended to AdS from Witten's proof of it.

Thus, when Jamie later writes:"This is as a negative value of Λ, if interpreted as vacuum energy, corresponds to a negative energy density (Hawking & Page 1982). This is a counterintuitive result, as although the negative masses are gravitationally repelling one another, the cosmological effect appears to be for the negative energy associated with negative masses to cause the universe to recollapse."it seems to me that the valid assignment of the label "counterintuitive result" is only to the claims by Jamie. Namely, if 1) what he states at the beginning of the paragraph is true, and indeed negative Λ universe can result from negative masses, AND 2) this happens in a way which is in agreement with Hawking&Page paper, that is the negative mass effect is seen only as negative energy density (...does positive mass theorem shall still apply?), and 3) this negative Λ results with recollapse of the universe, this last statement being less controversial per se (quoting Sean Carroll lectures: For Lambda < 0, the universe will always recollapse to a Big Crunch, either because there is a sufficiently high matter density or due to the eventual domination of the negative cosmological constant.), however, since all of this is happening because of 1) and 2) it also requires a closer look - the Big Crunch might not be a Big Crunch at all in the presences of all these repelling negative masses.

A spin-two field is a rank-2 tensor, a vector field is spin-1. Sorry in case the terminology is confusing. You find this explained in Weinberg's first QFT book (together will all other representations of the Lorentz-groups). With regards to gravity, you find the relevance of this explained both in Feynmann's lectures on gravity (he has a stretch where he goes on about anti-gravity) and in Zee's QFT Nutshell book, where he explains how the spin is relevant to tell you whether an interaction is attractive or repulsive.

A theory with fewer assumptions that explains the data as well as one with more assumptions is what we mean when we say a theory is "better." If you come up with a more complex theory (using your words - I don't think the reference to complexity is helpful here), you need to justify that by explaining more of the data. The trend that scientists, especially in the foundations, think that adding all kinds of unnecessary complications to their theories (supersymmetry, anyone?) actually improves the theories even though it does not help describing any observations is what's unscientific.

An attractive force between two bodies causes a change in their momenta toward each other, right? But a body with negative inertial mass will have the acceleration vector induced by a force point in the opposite direction from the force (and its momentum in the opposite direction of its velocity vector). So an attractive force (in the sense of change in its momentum) should repel a negative mass (in the sense of the change in its motion).

You write "General Relativity is a wonderful theory. It tells you how masses move under the pull of gravity. You do not get to choose how they move; it follows from Einstein’s equations. These equations tell you that like masses attract and unlike masses repel."

Perhaps I am being stupid, but it quite confuses me. I was under the impression that GR as a geometric theory would just fall apart if all particles did not respond identically to some 'gravitational pull', i.e., did not move along the very same geodesic (if they had the same initial position and velocity). If like masses attract, and unlike masses repel, how can, say, two particles with gravitational masses of opposite signs move along the same geodesic in the geometry of a particle with, say, positive gravitational mass? How can such a scenario at all be possible within GR? Please enlighten me :-).

Geodesics are a limit for (non-spinning) point particles. Point particles do (have to) respond identically to a gravitational field, as you say. But that limit breaks down if you have two masses of about equal value. (It also doesn't work if quantum effects become sizeable because then you're no longer dealing with points.) What kind of gravitational field you get in general, however, depends on the type of energy/mass you put into space-time. And note that since GR is not linear, it's not as simple as just flipping a sign. In general the solution for a negative source is just different from the solution for the same source with different sign, and they are not related in any simple way. That's most obvious in cosmology. The cosmological case without curvature for negative sources just doesn't have a (real valued) solution.

You said in the post that General Relativity mandates that like masses attract and unlike masses repel, but in your response to Pablo you said "in GR any type of mass will fall towards a positive mass object, including negative masses."

I suspect there's a nuance I'm missing - I'd be grateful for a clarification!

The connection with the anti-de Sitter (AdS) spacetime is weak. The AdS has a negative cosmological constant, Λ < 0, which means it has a source ~ Λg^{00} such that the quantum field that defines T^{00} is not bounded below. This means this vacuum can produce an endless amount of positive energy quantum fields, or for that matter positive vacuum bubbles that are de Sitter cosmologies such as what we are in, as the unstable negative vacuum locally plummets to greater negativity.. However, globally it does not change. The observable universe may then be some holographic screen with a positive energy junction on AdS_5. Positive dS spacetimes as holographic screens might then fit into a MERA tensor network of quantum information on AdS_5.

Farnes has a similar thing, but where negative mass-energy in halos falls into greater negative energy as positive matter is generated. This negative energy fluid or material would again not be bounded below quantum mechanically, and as such could generate a lot of positive energy by itself become more negative energy. The rate at which this would happen is an open question. However, I should think this would have observable consequences with early galaxies, say those with z > 1. The dynamics of a galaxy would change with time.

Both of these scenarios with AdS or Farnes' hypothesis have some similarity of Hoyle's continuous creation idea of matter spontaneously being generated in space. With the AdS case it would be vacuum bubbles of inflation that generate cosmologies. In Farnes' case is is almost more in line with Hoyle's idea.

If I assume that inertial mass and gravitational mass are identical in the EP, then we would have the rule of gravitation attraction or repulsion for the first the mass in F = ma and the second the attracting mass

+ + → attraction+ - → repulsion- + → attraction- - → repulsion

A negative mass halo would then tend to repel itself, but be attracted to the positive mass of the galaxy and the positive mass of the galaxy would be repelled. I would think this could generate an imploding positive mass core while the negative mass would expand away. I am not sure if this would reach some equilibrium. I think a numerical analysis would be needed to look at many cases to benchmark any analysis. It strikes me as complicated.

The main problem I see is that in the observable universe, with a positive Λ > 0 cosmological constant, the introduction of negative energy mass, which would presumably have a negative vacuum energy. This runs into a host of troubles from the existence of closed timelike geodesics to vacuum ambiguity.

The cosmological constant represents an energy density that is constant for all times. It can be eliminated in favor of a variable field energy density that would have to be included as a source term in the Einstein field equations. Then one can get on with the business of exploring the nature of the field; perhaps even quantizing it. Huseyin Yilmaz made a stab at this that needs to be reconsidered. It would be a good direction for a research effort whether Yilmaz got it right or not when he adopted Pauli's decomposition of the Einstein tensor in 1992.

Whatever the approach to negative masses in GR I don't think that the stability argument is a so serious one as one would "just" need to admit that gravity is not quantized as are the other interactions to eliminate most of the issue. Classical instabilities can be completely safe at the contrary to quantum instabilities.

What i think is the worst in J Farner approach is the fact that it leads to a k=-1 cosmology : i see no way how to make it compatible with BAOs, CMB first peak position and so on. And indeed completely ad hoc creation of matter is just an ugly idea. At least in the bimetric approaches , the creation could be a mere transfer of matter from one metric to the other which would make the idea much more acceptable ...

I think I have no problem understanding why the gravitational field for bodies with gravitational masses of opposite signs are in general very different. After all, loosely speaking, changing the sign of the mass, changes the sign of the energy-momentum tensor, which in turn changes positive curvature to negative curvature, and vice versa, i.e., spherical geometries to hyperbolic geometries, and vice versa.

But what happens in, say, the standard Schwarzschild spacetime if we place two (spinless) test particles (*) with gravitational masses of opposite signs? From your reply, I gather that they fall identically. But what then does the assertion "like masses attract and unlike masses repel" mean. I am still confused; please forgive me.

I have no qualifications whatsoever to comment on the science of this paper, or the above discussion. I do have a degree in biology, and some experience in life regarding How Stuff Works. The discussion of this paper did not pass the sniff test to me. It seemed to be more of a cludge that Made Stuff Work than a likely hypothesis. When something extrodinary makes multiple problems go away, the odds are against it working.

in Zee´s gravity book as Sabine said “depending on the spin” in chapter I.5 (17) “… we see that while like charges repel, masses attract.”But to my knowledge he does not explain how unlike masses repel, well, at least I never interpreted it this way. He continues: “We will see in chapter II.7 that the longitudinal degree of freedom of a massive spin 1 meson decouples as we take the mass to zero. The treatment given here for the interaction between charges (6) is correct. However, in the case of gravity, the 2/3 in (17) is replaced by 1 in Einstein’s theory, as we will see chapter VIII.1. Fortunately, the sign of the interaction given in (17) does not change.” (boldfaced by me)And in chapter VIII.1 he further reasons about the influence of a possible graviton mass: “… The paradox is formally resolved by noting that the higher orders are increasingly singular as” m_G→0. (The paradox relates to the “2/3 discontinuity”)So, maybe also get Feynman.

I believe there is an oversight in Sabine's thinking when she says "like masses attract, unlike masses repel". It is true that for like masses, the force vectors point towards each other and for unlike masses the force vectors point in the opposite direction. However, this does not mean the same is true for accelerations. For a negative mass, the acceleration is in the opposite direction to the force.

With this realisation, the exact dynamics described in Franes' paper occur, with two positive masses being accelerated towards each other as usual, two negative masses being accelerated apart from each other and a positive mass and a negative mass getting accelerated in a way so the negative mass chases the positive mass.

No conservation laws are broken in the low energy limit (like Newtonian dynamics plus Newtonian gravity) because of the unintuitive calculations with negative masses.

There's no such thing as a force in general relativity, and there are no inertial masses either. There's only a Lagrangian and Farnes has stated the Lagrangian at the beginning of the paper. What I am saying is simply this: You cannot start from a Lagrangian and then just postulate what you want to happen in the Newtonian limit. If what I say is wrong, fine, I will have learned something. I mean this sincerely - I'd be happy to hear if it works. But without a derivation I will not accept it.

Just a simple observation, but it would seem that a negative-matter halo surrounding each galaxy would need to stay clear of the positive-matter in the galaxy, or otherwise they would 'eat' each other like so many Pac-man icons on a computer screen. This would result in matter disappearing with no apparent cause, and no energy release as the opposing energy states of the two types of matter cancel each other out of existence. However, this is a rather precipitous comment, as I've only superficially scanned Jamie's paper, and there might be a mechanism therein to keep the two varieties of matter rigorously separated.

Lawrence crowell I'm wondering what makes you believe that " this would have observable consequences with early galaxies, say those with z > 1. The dynamics of a galaxy would change with time." why z>1 ?

Here is a fun exercise that I want all of you to try who don't understand why I say what I say. Take GR with a normal (massive) scalar field as source. Assume spherical symmetry. This system has solutions that correspond to a black hole surrounded by a scalar field - you find those in the literature if you don't want to solve the equations yourself (it's not difficult). No quantization implied - all classical. Now switch the sign in front of the energy term for the scalar field. Redo the calculation. Then try to bring the results in agreement with the idea that negative masses mutually repel.

Bee ,"But without a derivation I will not accept it." yet there are dozens of papers trying to reproduce a cosmological constant effect through a creation of matter modifying the effective equation of state of the global fluid ... most of those papers (exploiting an idea by Prigogin) are accepted in respected theoretical journals yet i rarely noticed any specific lagrangian describing the local physics of matter creation in most of them... more recently there have been attempts to write such lagrangians (everything in its own time) but it would not have been a good reason to refuse the earlier papers.

I also have a question about your own bimetric gravity theory : your pull overs are strange objects: they are not dynamical (in the sense that are not required to extremize your action) and seems to only be there to save the bianchi identities.

Indeed , if i look at your equations neglecting the effect of the "pullover fields" the first one predicts the binary pulsar inspiraling decay while the second one predicts the same binary pulsar gains more and more energy (by emitting negative energy gravitational waves) and is outspiraling.

So your pull over is crucial as it 's role is to reconcile the predictions of the two equations, so somehow a certain amount of energy is carried in or away from the system by the pull over field ... yet again it's not a dynamical field as i said above (not extremyzing the action)... that's so strange !are you sure that this game eventually allows you to have repelling gravity between the two sectors i.e that you can just set the pull over to a constant when solving for schwarzschild while a non constant pull over is at the contrary so crucial to consistently describe the binary pulsar decay ... my concern is that even the scharzschild solution ultimately relies on the exchange of gravitational waves..

"If you do this, you can produce particle pairs from a net energy of zero in infinitely large amounts. This fits badly with our observations."

Then how does the Big Bang follow? Would this not be more of a proof of a steady state universe, with elemental recycling through Black Hole, and SMBH producing fountains of ionic jets,sometimes forming structures like our own 'Fermi Bubbles', this hot plasma would cool and become atomic gasses, later coolng further to become molecular gasses, dust grains and larger agglomerates, one assumes that they would follow gravity and magnetic field lines back to the galactic disc, filling it with dust,, gasses cold enough to for stars, and then goes, eventually, though the grand recycle after the orbit decays and the matter finds itself, once again, feeding the SMBH. This would also be a basis for a continuing source for the Cosmic background Microwave Radiation and it's apparent bunchiness.

At least this is my perspective. Recent findings on the 'donut shape' of stellar ignition with jets and pulsars, BH's and SMH's all seem to form similar structures (as if fractal iterations) and that the gas movement, infall and feeding towards the black hole center would have an effect on the galactic rotation rates that would greatly lessen the need for dark matter in the balancing of equations of rotational curves in the galactic structure. Large volume, large scale galactic magnetic fields may 'herd' the cooling gas back into the proper 'feed' lanes and affect the speeds of the stars thus formed.

It seems as if an negative mass particle, upon encountering a photon, would accelerate itself back the direction the photon came from with equal force, and since photon is still trying to move forward, it will be constantly accelerated at C Backwards becoming either a 'Tachyon' or, perhaps the long-looked for 'graviton', a photon being accelerated backwards to it's source of origin, possibly being pushed towards the center of the mass originating the light ray/photon involved.

Just supposition, I am afraid. But may give you more, or different places to look and apply known theory. I may be just an arm-chair theorist, but it comes from a long and active life coupled to the sciences.

They're not dynamical fields for simplicity. Nothing preventing you from adding a dynamical term to the action to remedy that. They're not there "simply to save the Bianchi-identities" they are there for consistency as the Bianchi-identities, as the name says, are identities. You can't just throw them out, hence you can't throw the pull-overs out. They're frame-changes. I don't know what's strange about that. Two metrics, two ways to chose frames. If you don't keep track of the gauge-degrees of freedom, you get nonsense.

This is just what i meant : the pull over fields are on the right side because otherwise the equations could not be consistent :since the Bianchi identities are satisfied on the left side , as well the covariant derivative must vanish on the right side. I did not mean anything else.

Well , the theory you get when the pull over are fixed to the identity already in the action (but without duplicating somehow the mater action as you do) is mine ; it has a single equation instead of two, it does not have any Bianchi identities to satisfy because the geometrical part of the einstein equation is also modified. In general it has more functionnally independendent equations than GR: just means that it has more physical degrees of freedom : don't see any problem with that as long as the equations have solutions that reproduce in very good approximation the GR ones.

On the other hand i do agree that if you put the pull overs to the identity in your system of two equations you get nonsense...

I agree that you could add a kinetic term to your pull over. However my concern is that in your theory just as it is now, without any extra term you could have already treated your pull over as a dynamical object, because even if you call it and understand it as "frame-change" from the point of view of field theory , it's just a field as any other one and in general people believe that any field must extremize the action it belongs to.

Of course i suspect that if you do that you would get an additional equation spoiling the consistency of the theory.

So how do you justify not treating all objects in your theory (even your as well non dynamical other fixed "a" tensor ) as dynamical (in the sense that these should extremize the action even without adding any extra term) ?

I'm actually asking the question not because i think it's forbidden to do such things (i even found papers where people do the same for instance in unimodular gravity, but even in this case i found other papers in which theorists rather use Lagrange multipliers to fix a field and then still have actually everything extremizing the action ... why do they feel obliged to do that?) but because anytime i want to do the same , people juste refuse to hear what i have to say any longer ... and consider me as the perfect crackpot!

As I said, if you want to add a dynamical term, please go ahead. If you want to prove that makes the theory "inconsistent", please go ahead. Nothing stopping you from doing that. As you note, you can add the constraint with a Lagrange multiplier. I originally did this, but it seemed rather superfluous.

If JF is correct, then does this imply that a negative mass black hole can form? If not, why not, since these negative masses attract per JF? And if so, then what does the associated spacetime curvature look like? And can we completely neutralise a positive mass BH with a negative mass BH?

Right. The problem occurs if you suppose negative masses being inertial. They can be non-inertial virtual particles considered as net sum of - radiation quanta! But radiation quanta including all together (also neutrinos) is an open issue. Is it correlated with scalar field like Higgs or some projected wave energy as holographic information or composition? Many questions for daring researchers on this way of negative mass...

Summarum: negative mass seem to be rather a toy-model than an observable just like virtual particles generally are.

If your metrics are asymptotically Minkowskian (as you assume to get the Schwarzschild solution for instance) and if in the weak field case i write each of them as Minkowski plus small perturbation as usual, since the determinant of the pull over must be asymptotically equal to one, it also can be written as 1+ perturbation, and the pull overs themselves are Minkowski + small perturbation, or kronecker + perturbation (for those with both co and contra variant indices).

Then to first order all pull-over dependent terms in your equations dont contribute (can be set to one) and thus the two equations are not consistent in this case as the first one predicts an inspiraling decay of the binary pulsar while the second one predicts an outspiraling behaviour at the contrary.

Before i write this in details do you see something wrong with this perturbative approach in your theory.

Even if the metrics are not asymptotically equal but only up to normalisation factor, the conclusion is the same.

Hi Sabine, Can you please comment on the runaway motion effect when one adds negative mass in GR considering one single metric emerging from the Einstein field equations (hence in Jamie Farnes' theory, which has the same basis)? Indeed, Farnes writes in his paper, about the runaway motion:

"its behaviours violate no known physical laws. Negative masses are consistent with both conservation of momentum and conservation of energy"

But physicist Thomas Gold said otherwise, in the discussion about Bondi's paper with Peter Bergmann, Felix Pirani and Dennis Sciama in 1957:

"What happens if one attaches a negative and positive mass pair to the rim of a wheel? This is incompatible with general relativity, for the device gets more massive."

So the runaway motion seems to be clearly unphysical, as such perpetual motion machine would obviously violate conservation laws of physics. As a consequence, isn't Dr Farnes' toy model an overunity theory?

It seems that your own model introducing negative mass in cosmology ten years before Farnes (paper Bimetric theory with exchange symmetry published in Physical review D in 2008 but ignored today by the same media) is not plagued by such runaway motion paradox.

@ Frederic henry-couannier: The reason for using z > 1 or high z is this is the earlier universe and the abundance of this negative energy dark matter and positive matter would differ. This might mean galaxy dynamics would be different.

@Bee: The scalar field stress-energy tensor is

T_{ab} = (1/2m)G_{ab}^{cd}∂_cφ∂_dφ + (m/1)g_{ab}φ^2

where for m < 0 the T_{ab} < 0. This will violate the weak energy condition T^{00} ≥ 0. This also means the vacuum state is unstable with V(≥) < 0. I presume this is what you are referring to.

Farnes seems to have correctly described the straightforward generalization of Newtonian physics to negative masses, when inertial and gravitational mass are equal. So it seems that you are claiming that the Newtonian limit of GR to the case of objects with both negative inertial and gravitational mass is not in fact the usual Newtonian physics. Do we know what that limit is? Is it reasonable think there might be some alternative that does have the usual Newtonian physics in the Newtonian limit?

Negative masses and matter creation are more fantastic than dark matter and dark energy. Show me negative mass and matter creation in the lab or in telescope. Why do physicists make things up? I guess science fiction is more fun.

"Make theories as simple as possible, but not simpler." - Albert EinsteinIn other words, KISS (keep it simple stupid!)

Can't follow, sorry. You say asymptotically flat, perturbative approximation, then suddenly you have a binary inspiral. Where does that come from? What's spiraling? What are the sources? Also, why do you say the perturbation of the pull-overs don't contribute, I don't see that. That's a gauge-dependent statement and saying the metric asymptotically Minkowski doesn't fix the gauge.

I don't know how Farnes thinks the problem goes away, sorry. The explanation in the paper didn't make sense to me. The issue comes about for the reason that Lawrence noticed above, if you follow the logic of the paper (or that of Bondi's paper, respectively), you'll find that negative masses both repeal and attract positive masses. This clearly isn't good. You can try to work around this problem by saying either case occurs only in a certain limit, but then you will be left with the question what happens when one limit goes over into the other.

I used to think that the runaway problem is a consequence of the energetic instability that comes from having negative inertial masses. It's a decay to infinitely negative energies, basically, that becomes possible if you have no lower bound. I am not sure that's right, though.

Yes, the construction I have in my paper was carefully assembled to prevent those problems, both the runaway and the instability problem. As I point out somewhere (I believe in the appendix), I think that Bondi neglected to realize that if you have negative charges (in this case gravitational masses) you need a different covariant derivative acting on those charges. Hence the construction with the second metric and the second derivative, etc. Then you need a criterion to fix the second derivative, which is what the symmetry requirement is good for.

While I still think this does solve the cosmological constant problems (all except the coincidence problem), I gave up on the idea because I couldn't really see what else it was good for. The stuff doesn't make a good candidate neither for dark matter nor for dark energy (as I had originally hoped). Also, it's hideously difficult to find solutions to the system of equations. So, well. I don't know. Maybe in 100 years someone will find it useful for something.

That's one way to look at the instability, yes. You have to be careful though with specifying which stress-energy tensor you are talking about. There are two different ones, the canonical ones(the one obtained by variation with respect to the fields) and the gravitational one (that acts as a source in the field equations). Those two tensors are not generically identical.

As I already said above, there is no such thing as inertial mass in General Relativity. You write down a Lagrangian, there's a sign in front of the matter contribution - and that's that. That is all you have to play with as long as you stay in the confines of general relativity: A sign in the Lagrangian. Farnes writes down the Lagrangian and uses it in the first part of the paper, but then for the numerical simulation uses a separate set of equations without demonstrating that those are consistent.

As I said above, maybe Farnes is right, but without a derivation I will not accept it.

Typo in here: it should be “Zee´s QFT book”, but of course his gravity book is also excellent. Because of the “weak energy condition T^{00} ≥ 0” Lawrence just mentioned I never took the repelling case seriously.

"You say asymptotically flat, perturbative approximation, then suddenly you have a binary inspiral. Where does that come from? What's spiraling? What are the sources?"

OK , i just want to figure out what are the predictions of your theory as for the emission of gravitational waves by the Binary pulsar. So if i just consider the Binary pulsar in our side metric g i can neglect the matter on the other side metric hbar (just what you do when you apply your theory to the Schwarzschild case)...OK?

In this case your first field equation is just GR and from it you predict as GR that the binary is decaying, inspiraling because it loses energy emitted as gravitational waves ... OK ?

But then i look at your second equation and want to figure out what it means for the exact same system: if i could just set to one all pull-over terms in this second equation (as for your non dynamical fixed metric "a" , it plays no role here), then it would obviously predict that the same binary pulsar would at the contrary emit negative energy gravitational waves of the second metric hbar because the sign is not the same in the coupling between matter and the second metric... So the system would gain energy and be outspiraling. OK or do you need the equations for that, which i can send to you by email because you would not accept a link to my website here?

So the pull over must of course be crucial in your theory otherwise the predictions of the 2 equations are conflicting and the theory is not consistent...of course we agreed on that from the begining but here i just wanted to make it appearant in the concrete case of the binary pulsar and not a more formal mathematical argument...

Now the problem is that actually it seems that the perturbative approach fails for your theory (and it is the perturbative approach that is always used to treat the binary pulsar in GR). Because if i not only write the metric as minkowski plus perturbation but also all the Pull-over terms (then minkowski or 1 or kronecker + perturbation depending on the specific pull-over object), then the leading order term on the source side is just the matter tensor only while pull-over perturbation times matter tensor is next to leading order.

so i'm not saying that the pull-overs can be set to 1 exactly but only that their leading contribution is 1 (in a perturbative approach just because the pull-overs are asymptotically one or at least a constant, it does not matter, as usual the dynamics of the background is completely negligible when treating gravitational waves)... and then i'm back to the problem mentionned above: to leading order your two field equations would have conflicting predictions!

So somehow it appears and i'm afraid that the pull-overs can't be treated perturbatively... what it means and if it is or not a real concern is the question i was asking to you...my impression is that the pull-overs can't do their job to reconcile the predictions of the two equations because there is actually no pull-over field (at least one that could be treated perturbatively) existing solution to that problem ...

lawrence: The AdS has a negative cosmological constant, Λ < 0, which means it has a source ~ Λg^{00} such that the quantum field that defines T^{00} is not bounded below. This means this vacuum can produce an endless amount of positive energy quantum fields

Just to be sure... a cosmological constant is a nondynamical staff, it's not related to a field needing to be quantized ... so what you say applies to J Farnes model in which the negative cosmological constant is an effective description but would not apply to a true negative cosmological constant right ?

First, of course the pull-overs are crucial. That's why they are there. Second, you can treat gravitational waves perturbatively but not the sources. Third, you are mixing up different types of energies. The objects in the system do not gain kinetic energy through the emission of gravitational waves in either which case because the equivalence principle is violated.

this is way off-topic but i saw your tweet (i don't do social media so your posting tweets on your blog is vital to my keeping up with physics related material) about your twins being in a play. why no pics of them in the play?

To begin with, the play hasn't taken place yet, it's a Christmas story! Besides this, I wouldn't publicly share picture with other people's children. If I share pictures of my own kids, I do it mostly on facebook under a privacy setting - sorry about that. But we'll probably have a family picture for the Christmas cards, which I can upload to the blog, in case you want to see how we have all gotten older :o)

Sabine, it's first worth mentioning that Hermann Bondi, the cosmologist who is credited with developing much of the mathematics of gravitational waves at a time when it was controversial whether they were a prediction of general relativity and has the Bondi k-calculus and the Bondi mass named after him, derived the same dynamics as in Franes' paper.

My own reasoning about dynamics is based on extending a Newtonian dynamics and Newtonian gravity to the case where negative masses are permitted (and inertial mass is assumed proportional to gravitational mass). There is only one way to do this which respects the basic conservation laws. The forces between masses are as you describe: for two positive masses the forces point inward. For two negative masses, the forces point inward, for a positive mass and a negative mass, the forces point outward. But the accelerations for negative masses are in the opposite directions to those forces. This is essential for conservation of momentum. This results in using exactly the same maths where some of the masses are negative. [I also extended this to a special relativistic scenario in the exotic case of a negative mass chasing a positive mass. Dilation ensures that the acceleration only makes the velocity asymptotic to c in the case where the total mass is zero and the relative velocity is zero. This was not clear to me in Franes' paper).

It seems to me there cannot be any other low speed, low escape speed approximation to general relativity with negative masses than the above, which would imply that the dynamics described by Bondi and Franes are correct when speeds and gravitational fields are small. Clearly if the signs are the way they are in this domain, they can't be the opposite in general.

I am not sure of your reasoning that the dynamics is not that of Bondi/Franes, but I suspect it is the result of an assumption that is always true for positive masses but would be flipped for negative masses. Could you provide a reference?

Ma'am, Another thought just crossed my mind about the detection of such negative mass particles or objects is this: How would tidal tails from colliding or near-colliding galaxies be affected? Should we see evidence in the tidal tail shapes, or is that one way to be able to disprove the paper concerning negative mass particles.

On the other hand, the possibility of an anti-mass field would be an area to probe, even in the absence of a discrete small particle, perhaps a larger very dispersed field would be of similar effect.

I personally have troubles with the gravity lensing used for locating dark mass presently, as I do not feel it takes into consideration the normal refraction properties of the gas/plasma around many of the galactic clusters that are used to highlight the effect, may have some bearing on whether there is really more (dark) matter needed for spacetime bending, or if the gas is refracting like a lens on it's own. Spherical and lobed gas/plasma masses should refract light at various wavelengths without having to resort to a DM explanation, or at least greatly diminish the need for it in equations. Negative mass or anti-mass would seem to have no bearing on such curves.

Just a minor correction: you say Farnes gives a Lagrangian at the beginning of the paper. As far as I can tell, he only gives the Einstein Field Equation and the Friedmann equations (which are just the EFE specialized to the case of a homogeneous, isotropic universe). (Also, he doesn't appear to use the second Friedmann equation at all in the remainder of the paper, which I find to be a key omission.) This doesn't change the substance of your criticism that there is no consistency check between the "GR" part of the paper and the "Newtonian" part.

The AdS vacuum does not need a non-gravitational source. It is not hard to see how the near horizon condition of a black hole for an accelerated frame is AdS_2xS^2. This is in a region that is a pure vacuum as black holes have pure vacuum. However we know that black hole emit Hawking radiation and there is a metric back reaction. The near horizon condition for an accelerated observer has negative curvature, and this generates particles. Another way to think of it is the accelerated observer has time accelerated so the observer encounters a lot of Hawking radiation pouring out. In a similar way the AdS_5 could be the perfect source for the generation of nascent cosmologies.

@ Bee: I am aware of there being different T^{ab}s. I was thinking of the stress-energy such as presented in Ellis and Hawking. BTW the G_{cd} is a lot of metric stuff that I did not want to write down.

Any inspiral that occurs with galaxies, here now not considering negative masses, is due to friction. Gravitational radiation has a term ~ GQ/c^5r^4, where Q is a quadrupole moment and r distance. This means gravitational radiation has an acceleration on the order of GMd^2/c^5r^4, for d the distance between two test masses. While M may involve 10^{11} stars, each at around 10^{30}kg the radial distance is on the order of 10^5 light years or 10^{18}m. So just run a numerical order of magnitude calculation and find the acceleration between stars at opposite ends of the galaxy would be at most 10^{-36}m/s^2 or so. So for all the mass in a galaxy they are not a big emitter of gravitational radiation.

Galaxies may contract due to friction, say by stars running through dust and gas. This is in fact what sets up density waves in spiral arms, as I understand.

The push-pull of positive and negative masses may have some dynamical effect if this is the case. With the generation of positive mass this might lead to some detailed balance questions on how galaxies are in fact comparatively stable.

The gravitational field for a black hole in GR is a vacuum solution. (That's because the singularity is not part of space-time.) The mass is an integration parameter. You can chose it to be a negative number, if you wish, no problem. But the resulting space-time is no longer a black hole. It would repel all normally gravitating masses, including light, hence no horizon.

I've read more of Jamie Farne's paper, which has a clarity of language which makes it a rather enjoyable read. In the sections that I've read there's no mention of any (EM) radiation coupling between this negative-energy matter and our normal, positive-energy matter, though it's possible I overlooked an area where he addresses that. Now, the simplest extrapolation of the identity of the negative-energy matter in Jamie's model would be negative-energy versions of our Standard Model particles. To impute anything else would require an additional free parameter, or parameters, increasing the model's complexity, so less desirable.

Assuming that the Dark Matter halos around galaxies consist of a plasma of negative-energy electrons and protons, that mostly haven't coalesced to form atoms due to the gravitational repulsion, then it would follow that the acceleration of these charged particles would result in the emission of EM radiation. Being that the source of this radiation possesses negative energy, the radiation would, I assume, possess a backwards-in-time direction. This is reminiscent of of John Cramer's Transactional model which implicitly requires backwards-in-time radiation.

Presumably, the absorption of such radiation by normal atoms would cause them to lose energy - the reverse of what occurs when a normal photon interacts with an atom. As per the assumption that Jamie's Dark Matter is normal matter with an inverted energy sign, this could be a way to confirm, or refute, his hypothesis.

I'm just an average Joe construction worker reading through this discussion. I must say this is over my head but I find it fascinating and I wish I understood more of it. I have been interested in the concept of this elusive matter and energy but that's as far as it goes. I was wondering how time fits into the picture. In my mind time is created by matter. If time does not exist at all without matter then could it have an effect on things proportional to their Mass? Like photons not having much mass would be less subject to time? Anyway just daydreaming about it and wondering about the need for a constantly replenishing energy source that's invisible. In my mind time is something that is constantly created but yet does not have physical properties. Could time itself be the propellant accelerating the expansion of the universe as more and more comes into existence? No I'm not on drugs :-)

It's debatable whether a cosmological constant counts as source or not, but imo it's not a very useful debate. What I meant to say is that if you know the constant is constant you can solve the equation right away.

"I also don't see energy conservation being used in lieu of the second Friedmann equation; it seems to me that one would still need to use at least one of them."

What I tried to say is that rather than using both Friedmann equations, you can use one of them plus energy conservation. In the case of the cc, you have basically assumed energy conservation by postulating that it's constant, hence you're done with integrating the first. Just look at the equation: it's a first-order dgl with only constants in it. You can solve it without drawing on another equation.

But it is still unclear what exactly he means when he says that constant negative mass density is similar to negative cosmological constant. He uses the first Friedman equation, but I don't see how he can claim that. And from the second equation it is obvious that negative mass density is similar to a positive cosmological constant.

Bee wrote something I thought of writing and am going to comment a bit in agreement. The Einstein field equation tells us that the traceless part of the Ricci tensor R_{ab} - 1/2Rg_{ab} equals the stress-energy tensor times a very small coupling constant 8πG/c^4 T_{ab}. To make things a bit confusing Einstein found that his equations predicted the universe, then thought to be just the Milky Way with other galaxies thought to be just nebula, would collapse. Einstein then added a term -Λg_{ab} to counter this implosion. If there is no matter or non-gravity field sources so T_{ab} = 0 then we have the Einstein field equation that the Ricci tensor is equal to various terms times the metric. This is an Einstein space.

This was found not to be quite right with Edwin Hubble's find in 1927 that some nebula were galaxies and were receding away. Einstein called this his biggest blunder, but really it was of considerable insight. The cosmological constant might be though of as -Λg_{ab} = 8πG/c^4 T_{ab} for some source of a constant energy density and pressure in the spacetime. This stress energy is then something of the form T_{ab} = ⟨Ω|T_{ab}|Ω⟩ for T_{ab} the quantum operator corresponding to the stress-energy of a quantum field. However |Ω⟩ is a vacuum state and so we tend to think of this as a vacuum solution with no source. It is a matter of definition.

There are some subtle aspects to this, and we might consider the Ricci flow equation of Hamilton

∂_tg_{ij} = -2R_{ij} + (2/d) ⟨R_{ij}⟩

for the last term pertaining to a compact manifold of dimension d and ⟨R_{ij}⟩ the average of the Ricci tensor, with a suggestive quantum expectation notation. This was featured prominently in Perelman’s proof of the Poincare conjecture. In a complex valued setting it is not hard to imagine this flow equation as having some stationary condition with ∂_tg_{ij} a constant times the metric. Since this equation in two dimensions has conformal properties we might imagine this as a 4 dim manifold in some variation with some additional time(like) variable. This for AdS would give some correspondence between the negative cosmological constant, the additional time variable and quantum expectations. The 4-dim manifold would then uphold the Cardy equations and so forth.

The question over whether the cosmological constant term is really a vacuum term, which in the Einstein field equations really refers to a classical vacuum of absolute nothing, or whether we can sneak quantum vacuum stuff into the idea underlies some aspects of our current lack of understanding on these matters.

No experiment has ever seen evidence of negative masses of any kind, at least not of the fundamental kind. There is something called a "negative effective mass" in condensed matter systems, which is (loosely speaking) defined relative to a background medium and is not indeed negative.

if you know the constant is constant you can solve the equation right away

Yes, I see that, but in the proposed model where "negative mass" is supposed to act like a negative cosmological constant, the density of "negative mass" is not necessarily constant, is it? If it isn't, then there are dynamics that need to be solved, and I don't see that being done in the paper.

I don’t understand why it is assumed that black holes evaporate. Most of the energy that they absorb from Hawking radiation comes in the form of photon pairs. The negative energy photon is a just like a positive energy photon except it has a negative frequency. Its mass is the same as the positive energy photon. Any black hole get bigger as time goes on as it extracts energy from the vacuum.

This extraction of energy from the vacuum is how the vacuum gets more unbalanced as time goes on. This growing energy imbalance in the vacuum is where eternal inflation comes from.

The amount of Hawking radiation generated by black holes is approximately constant so too is the expansion rate of the universe.

A black hole sitting in a space with a constant vacuum energy density might at first blush be thought to increase mass by frame dragging all this vacuum energy into it. That would in many ways be very troubling. However the condition that pressure p = wρ, for ρ the vacuum energy density and with w = -1, means the net energy gain by a black hole sitting in space is zero. This is a theoretical or phenomenological argument in favor of the w = -1 model of the universe. Current observation are close to this value.

When cosmologists speak of "fluids" they normally mean this as an effective bulk description of a collection of particles (I know it's confusing, but a fluid can be a gas). In principle this particle could be detected, but whether or not that's possible depends on how strong the interaction is. If it interacts very rarely, it might take a million years to catch one or such. You cannot tell how strongly it interacts without writing down a microscopic theory. In summary, there's a priori no obvious contradiction between Farnes' idea and existing measurements. Best,

Maybe one day someone will write a microscopic theory of DM, one that does not depend on the Standard Model ... perhaps they will also develop a microscopic model of Farnes' negative mass fluid, one that likewise does not depend on the Standard Model.

Apart from the fact that Farnes' idea is much newer, attempts to describe what DM might be, particle-wise, and that tests of how well it matches observations have also not been done yet, is there any significant difference?

It might be worth noting what kinds of observation DM can explain, in terms of the astrophysics; here's an initial, very shorthand, list: the CMB (acoustic peak), lensing, galaxy rotation curves, galaxy cluster dynamics. From observations, the minimum size of "DM clumps" is some tens of parsecs (YMMV).

Dust and gas can be transformed into a Super fluid by a process of quantum mechanical entanglement with photons, plasmons, and phonons in which cold plasma forms. This Bose Einstein condensate can extend for cosmological distances and fill up the space in and between galaxies. This condensate can exist in a “Dark” state where it absorbs energy and does not radiate any to the far field.

https://phys.org/news/2018-10-bose-einstein-condensate-space.html"A Bose-Einstein condensate has been produced in space for the first time"

I observe "dark" phenomena in my mathematical model. "Dark" phenomena exist, but the "dark" name is unfortunate is no good for combine all the properties of these phenomena.

Math GR is not perfect. Mathematics GR is not a criterion of truth.

The error of the article under discussion, the error of your criticism, the error of GR and the correct solution of the problem are in the following area:

We formulated a postulate on the independence of the speed of light from the speeds of the source and receiver of light, for example. We do not have mathematics to describe independent relations of this type.

The projection of Einstein's intuitive model onto existing mathematics is weaker than this intuitive model.

We can use imaginary unit here, for example. We can explore some properties of independent relationships with help of the imaginary unit, but this tool is not perfect.

Existing mathematics is a brake on physics and other branches of science and culture. "The mathematical map of material reality differs from the territory" - this statement is correct, but it is correct for existing mathematics only.

Existing mathematics is not a criterion of truth. The ideas of physicists may be more correct than the mathematical projections of these ideas. I do not evaluate the article under discussion. The solution is in another area. The solution is at paradigm level.

Scientists use an arbitrary choice of basic postulates from a variety of possible postulates. We can solve the problem of an arbitrary start with the help of new type of mathematics.

We can build non-rational mathematics. A mathematical map can be equivalent territory if the mathematics is non-rational. Rational mathematics is a fragment of non-rational mathematics. The path to self-sufficient mathematics lies through a discussion of the paradigm.

I have such math as a fact. Collective science lives in a different paradigm. The collective path to mathematics lies through the collective creation of this mathematics. Publishing mathematics with a critical analysis of the existing solution does not work in this case. Mathematics proves itself. Scientists cannot destroy self-sufficient mathematics through critical analysis. However, the new math consist contradictions; contradiction is an error criterion and does not make sense for the current paradigm. The path to the solution lies through the analysis of the paradigm.

A superfluid does not form by "a process of quantum mechanical entanglement with photons, plasmons, and phonons in which cold plasma forms". To begin with a superfluid isn't a plasma. Also, it doesn't come about by entanglement with photons. Plasmons and phonons are quasi particles, the latter of which can exist once a superfluid has formed, but it does not cause the formation. I would really appreciate if you stopped posting ill-informed comments about topics you clearly do not understand.

@Axil: it was meant as a joke, hence the ;-) Yeah, I know, everyone tells me my jokes are not funny. I should stop trying to make them.

As Sabine has noted, GR says nothing about things like quarks, photons, atoms, even fluids (liquids and gases) and dust. In the everyday sense of those words. However, both fluid and dust are used in GR, where they have quite narrow meanings; there’s a Wikipedia article on this.

In Einstein's GRT the motion of a small positive mass point test particle is ruled by the geodesic equation in curved 4-dimensional space-time. But what I believe I have learned from my friend and GRT expert Heinz Dehnen (Ret. U. Konstanz),the motion of a small negative mass point test particle would be ruled by an anti-geodesic, which together with geodesics is not possible in a Riemannian space-time. Please correct me if I am wrong.F. Winterberg (Ret. U. Nevada, Carl Friedrich Gauss Academy of Science, Potsdam Germany)[fwinterbe@gmail.com].

The mistake is in this part of the wording, for example. We know the properties of 4-dimensional models from mathematics, and we do not observe these properties in reality. If we observe the properties of another model, then we must use the mathematical writing of this model. The diagnosis is as follows: We do not have (published) mathematics for this.